A PBX is a Private Branch Exchange, which is sort of like having a switch or central office located at a customer's premises (on or about a customer's physical location). In a circuit-based system, a PBX would provide various functional aspects such as routing calls and providing connectivity to the outside world. But they are often very expensive and can cost in the hundreds of thousands of dollars to purchase, install, etc.
But an analog PBX designed to be operated in a circuit-based network will not function in a packet-based network. VOP (voice over packet) is a term associated with a packet-based network environment that allows voice communications to occur utilizing a packet-based rather than a switch-based network.
Customers can reap benefits from using VOP technology to make and receive telephone calls because providing telephone service via VOP is often less expensive than the traditional service that utilizes the PSTN and is heavily regulated. But in such a setting, a customer may want features that have traditionally been offered in a PSTN environment via a PBX, but that are not available in a VOP environment because of the stark technological differences between a packet-based environment (VOP) and a circuit-based environment (PSTN). Moreover, customers may desire such functionality without having to buy a PBX, even one that might be able to work in a packet-based environment; that is, on the customer side—between the PBX and all the individual endpoints—rather than on the network side.